Guaranteed sensor coverage with the weighted-$D^2$ sampling

In this paper we focus on the mobile sensor coverage problem formulated as a continuous locational optimization problem. Cort\`es et al. first proposed a distributed version of the Lloyd descent algorithm with guaranteed convergence to a local optima. Since then researchers have studied a number of variations of the coverage problem. The quality of the final solution with the Lloyd descent depends on the initial sensor configuration. Inspired by the recent results on a related $k$-means problem, in this paper we propose the weighted-$D^2$ sampling to choose the initial sensor configuration and show that it yields $O(\log k)$-competitive sensor coverage before even applying the Lloyd descent. Through extensive numerical simulations, we show that the initial coverage with the weighted-$D^2$ sampling is significantly lower than that with the uniform random initial sensor configuration. We also show that the average distance traveled by the sensors to reach the final configuration through the Lloyd descent is also significantly lower than that with the uniform random configuration. This also implies considerable savings in the energy spent by the sensors during motion and faster convergence.